Why Do Life Insurance Policyholders Lapse? The Roles of Income, Health and Bequest Motive Shocks

Transcription

1 Why Do Life Insurance Policyholders Lapse? The Roles of Income, Health and Bequest Motive Shocks Hanming Fang Edward Kung May 2, 2010 Abstract We present an empirical dynamic discrete choice model of life insurance decisions designed to bypass data limitations where researchers only observe whether an individual has made a new life insurance decision but but do not observe the actual policy choice or the choice set from which the policy is selected. The model also incorporates serially correlated unobservable state variables, for which we provide ample evidence that they are required to explain some key features in the data. We empirically implement the model using the limited life insurance holding information from the Health and Retirement Study (HRS) data. We deal with serially correlated unobserved state variables using posterior distributions of the unobservables simulated from Sequential Monte Carlo (SMC) methods. Counterfactual simulations using the estimates of our model suggest that a large fraction of life insurance lapsations are driven by i.i.d choice specific shocks, particularly when policyholders are relatively young. But as the remaining policyholders get older, the role of such i.i.d. shocks gets less important, and more of their lapsations are driven either by income, health or bequest motive shocks. Income and health shocks are relatively more important than bequest motive shocks in explaining lapsations when policyholders are young, but as they age, the bequest motive shocks play a more important role. Keywords: Life insurance lapsations, Sequential Monte Carlo Method JEL Classification Codes: G22, L11 Preliminary and Incomplete. All comments are welcome. We have received helpful comments and suggestions from Han Hong, Aprajit Mahajan, Jim Poterba and Ken Wolpin. Fang would also like to gratefully acknowledge the generous financial support from the National Science Foundation through Grant SES All remaining errors are our own. Department of Economics, University of Pennsylvania, 3718 Locust Walk, Philadelphia, PA 19104; Duke University and the NBER. hanming.fang@econ.upenn.edu Department of Economics, Duke University, 213 Social Sciences Building, P.O. Box 90097, Durham, NC edward.kung@duke.edu

2 1 Introduction The life insurance market is large and important. Policyholders purchase life insurance to protect their dependents against financial hardship when the insured person, the policyholder, dies. According to Life Insurance Marketing and Research Association International (LIMRA International), 78 percent of American families owned some type of life insurance in By the end of 2008, the total number of individual life insurance policies in force in the United States stood at about 156 million; and the total individual policy face amount in force reached over 10 trillion dollars (see American Council of Life Insurers (2009, p )). There are two main types of individual life insurance products, Term Life Insurance and Whole Life Insurance. 1 A term life insurance policy covers a person for a specific duration at a fixed or variable premium for each year. If the person dies during the coverage period, the life insurance company pays the face amount of the policy to his/her beneficiaries, provided that the premium payment has never lapsed. The most popular type of term life insurance has a fixed premium during the coverage period and is called Level Term Life Insurance. A whole life insurance policy, on the other hand, covers a person s entire life, usually at a fixed premium. In the United States at year-end 2008, 54 percent of all life insurance policies in force is Term Life insurance. Of the new individual life insurance policies purchased in 2008, 43 percent, or 4 million policies, were term insurance, totaling $1.3 trillion, or 73 percent, of the individual life face amount issued (see American Council of Life Insurers (2009, p )). Besides the difference in the period of coverage, term and whole life insurance policies also differ in the amount of cash surrender value (CSV) received if the policyholder surrenders the policy to the insurance company before the end of the coverage period. For term life insurance, the CSV is zero; for whole life insurance, the CSV is typically positive and pre-specified to depend on the length of time that the policyholder has owned the policy. One important feature of the CSV on whole life policies relevant to our discussions below is that by government regulation, CSVs does not depend on the health status of the policyholder when surrendering the policy. 2 Lapsation is an important phenomenon in life insurance markets. Both LIMRA and Society of Actuaries considers that a policy lapses if its premium is not paid by the end of a specified time (often called the grace period). This implies that if a policyholder surrenders his/her policy for 1 The Whole Life Insurance has several variations such as Universal Life (UL) and Variable Life (VL) and Variable- Universal Life (VUL). Universal Life allows varying premium amounts subject to a certain minimum and maximum. For Variable Life, the death benefit varies with the performance of a portfolio of investments chosen by the policyholder. Variable-Universal Life combines the flexible premium options of UL with the varied investment option of VL (see Gilbert and Schultz, 1994). 2 The life insurance industry typically thinks of the CSV from the whole life insurance as a form of tax-advantaged investment instrument (see Gilbert and Schultz, 1994). 1

3 By Face Amount By Number of Policies Table 1: Lapstion Rates of Individual Life Insurance Policies, Calculated by Face Amount and by Number of Policies: Source: American Council of Life Insurers (2009) cash surrender value, it is also considered as a lapsation. According to Life Insurance Marketing and Research Association, International (2009, p. 11), the life insurance industry calculates the annualized lapsation rate as follows: Number of Policies Lapsed During the Year Annualized Policy Lapse Rate = 100 Number of Policies Exposed to Lapse During the Year. The number of policies exposed to lapse is based on the length of time the policy is exposed to the risk of lapsation during the year. Termination of policies due to death, maturity, or conversion are not included in the number of policies lapsing and contribute to the exposure for only the fraction of the policy year they were in force. Table 1 provides the lapsation rates of individual life insurance policies, calculated according to the above formula, both according to face amount and the number of policies for the period of Of course, the lapsation rates also differ significantly by age of the policies. For example, Life Insurance Marketing and Research Association, International (2009, p. 18) showed that the lapsation rates are about 2-4% per year for policies that have been in force for more than 11 years in Our interest in the empirical question of why life insurance policyholders lapse their policies is primarily driven by recent theoretical research on the effect of the life settlement market on consumer welfare. A life settlement is a financial transaction in which a policyholder sells his/her life insurance policy to a third party the life settlement firm for more than the cash value offered by the policy itself. The life settlement firm subsequently assumes responsibility for all future premium payments to the life insurance company, and becomes the new beneficiary of the life insurance policy if the original policyholder dies within the coverage period. 3 The life settlement industry is quite recent, growing from just a few billion dollars in the late 1990s to about $12-$15 billion in 2007, and according to some projections (made prior to the 2008 financial 3 The legal basis for the life settlement market seems to be the Supreme Court ruling in Grigsby v. Russell [222 U.S. 149, 1911], which upheld that for life insurance an insurable interest only needs to be established at the time the policy becomes effective, but does not have to exist at the time the loss occurs. The life insurance industry has typically included a two-year contestability period during which transfer of the life insurance policy will void the insurance. 2

4 crisis), is expected to grow to more than $150 billion in the next decade (see Chandik, 2008). 4 In recent theoretical research, Daily, Hendel and Lizzeri (2008) and Fang and Kung (2010a) showed that, if policyholders lapsation is driven by their loss of bequest motives, then consumer welfare is unambiguously lower with life settlement market than without. However, Fang and Kung (2010b) showed that if policyholders lapsation is driven by income or liquidity shocks, then life settlement may potentially improve consumer welfare. The reason for the difference in the welfare result is as follows. Life insurance is typically a long-term contract with one-sided commitment in which the life insurance companies commit to a typically constant stream of premium payments whereas the policyholder can lapse anytime. Because the premium profile is typically constant, the contracts are typically front-loaded, that is, in the early part of the policy period, the premium payments exceed the actuarially fair value of the risk insured. In the later part of the policy period, the premium payments are less than the actuarially fair value. As a result, whenever a policyholder lapses his/her policy after holding it for several periods, the life insurance company pockets the so-called lapsation profits, which is factored into the pricing of the life insurance policy to start with due to competition. The key effect of the settlement firms on the life insurers is that the settlement firms will effectively take away the lapsation profits, forcing the life insurers to adjust the policy premiums and possibly the whole structure of the life insurance policy under the consideration that lapsation profits do not exist. In the theoretical analysis, we show that life insurers may respond to the threat of life settlement by limiting the degree of reclassification risk insurance, which certainly reduces consumer welfare. However, the settlement firms are providing cash payments to policyholders when the policies are sold to the life settlement firms. The welfare loss from the reduction in extent of reclassification risk insurance has to be balanced against the welfare gain to the consumers when they receive payments from the settlement firms when their policies are sold. If policyholders sell their policies due to income shocks, then the cash payments are received at a time of high marginal utility of income, and the balance of the two effects may result in a net welfare gain for the policyholders. If policyholders sell their policies as a result of losing bequest motives, the balance of the two effects on net result in a welfare loss. Thus, to inform policy-makers on how the emerging life settlement market should be regulated, an empirical understanding of why policyholders lapse is of crucial importance. For this purpose, we present and empirically implement a dynamic discrete choice model of life insurance decisions. The model is semi-structural and is designed to bypass data limitations 4 The life settlement industry actively targets wealthy seniors 65 years of age and older with life expectancies from 2 to up to years. This differs from the earlier viatical settlement market developed during the 1980s in response to the AIDS crisis, which targeted persons in the age band diagnosed with AIDS with life expectancy of 24 months or less. The viatical market largely evaporated after medical advances dramatically prolonged the life expectancy of an AIDS diagnosis. 3

5 where researchers only observe whether an individual has made a new life insurance decision (i.e., purchased a new policy, or added to/changed an existing policy) but do not observe what the actual policy choice is nor the choice set from which the new policy is selected. We empirically implement the model using the limited life insurance holding information from the Health and Retirement Study (HRS) data. An important feature of our model is the incorporation of serially correlated unobservable state variables. In our empirical analysis, we show ample evidence that such serially correlated unobservable state variables are important for explaining some key features in the data. Methodologically, we deal with serially correlated unobserved state variables using posterior distributions of the unobservables simulated from Sequential Monte Carlo (SMC) methods. 5 Relative to the few existing papers in the economics literature that have used similar SMC methods, our paper is, to the best of our knowledge, the first to incorporate multi-dimensional serially correlated unobserved state variables. In order to give the three unobservable state variables in our empirical model their desired interpretations as unobserved income, health and bequest motive shocks, this paper proposes two channels through which we can anchor these unobservables to their related observable variables. Our estimates for the model with serially correlated unobservable state variables are sensible and yield implications about individuals life insurance decisions consistent with the both intuition and existing empirical results. In a series of counterfactual simulations reported in Table 15, we find that a large fraction of life insurance lapsations are driven by i.i.d choice specific shocks, particularly when policyholders are relatively young. But as the remaining policyholders get older, the role of such i.i.d. shocks gets less important, and more of their lapsations are driven either by income, health or bequest motive shocks. Income and health shocks are relatively more important than bequest motive shocks in explaining lapsation when policyholders are young, but as they age, the bequest motive shocks play a more important role. The remainder of the paper is structured as follows. In section 2 we describe the data set used in our empirical analysis and how we constructed key variables, and we also provide the descriptive statistics. In Section 3 we describe the empirical model of life insurance decisions. In section 4 we provide some preliminary reduced-form and static structural models describing the relationship between life insurance decisions and individuals observable characteristics. In section 5 we expand the model to explicitly account for dynamics, and we use counterfactual simulations to investigate the effects of income and bequest shocks to lapsation patterns. In section 5 See also Norets (2009) which develops a Bayesian Markov Chain Monte Carlo procedure for inference in dynamic discrete choice models with serially correlated unobserved state variables. Kasahara and Shimotsu (2009) and Hu and Shum (2009) present the identification results for dynamic discrete choice models with serially correlated unobservable state variables. 4

6 6 we extend the dynamic model to include serially correlated unobserved state variables, and describe a method to account for them in estimation. In Section 7 we report our counterfactual experiments using the model with unobservables. In section 8 we conclude. 2 Data We use data from the Health and Retirement Study (HRS). The HRS is a nationally representative longitudinal survey of older Americans which began in 1992 and has been conducted every two years thereafter. The HRS is particularly well suited for our study for two reasons. First, the HRS contains rich information about income, health, family structure, and life insurance ownership. If family structure can be interpreted as a measure of bequest motive, then we have all the key factors motivating our analysis. Second, the HRS respondents are generally quite old: between 50 to 70 years of age in their first interview. As we described in the introduction, the life settlements industry typically targets policyholders in this age range or older, so it is precisely the lapsation behavior of this group that we are most interested in. Our original sample consists of 4,512 male respondents who were successfully interviewed in both the 1994 and 1996 HRS waves, and who were between the ages of 50 and 70 in We chose 1996 as the period to begin decision modeling because starting in 1996 the HRS began to ask questions about whether or not the respondent lapsed any life insurance policies and whether or not the respondent obtained any new life insurance policies since the last interview. These questions are used in the construction of the decision variable in the structural model. We use only respondents who were also interviewed in 1994 so we can know whether or not they owned life insurance in We follow these respondents until Any respondent who missed an interview for any reason other than death between 1996 and 2006 was dropped from the sample. Any respondent with a missing value on life insurance ownership any time during this period was also dropped. This leaves us a sample of 3,567 males. We also dropped 243 individuals who never reported owning life insurance during all the waves of HRS data. Our final analysis sample thus consists of 3,324 in wave 1996 and the survivors among them in subsequent waves, 3,195 in wave 1998, 3022 in wave 2000, 2,854 in wave 2002, 2,717 in wave 2004 and 2,558 in wave Table 2 describes how we come to our final estimation sample. Construction of Variables Related to Life Insurance Decisions. Except for the variables related to life insurance decision, all other variables used in our analysis are taken directly from the RAND distribution of the HRS. Here we describe the questions in HRS we use to construct the 5

7 Table 2: Sample Selection Criterion and Sample Size Selection Criterion Sample Size All individuals who responded to both 1994 and 1996 HRS interviews 17, males who were aged between 50 and 70 in 1996 (wave 3) 4, did not having any missing interviews from 1994 to , did not have any missing values for reported life insurance ownership status from 1994 to , reported owning life insurance at least once from 1996 to ,324 Note: The selection criteria are cumulative. life-insurance related variables. For whether or not an individual owned life insurance in the current wave, we use the individual s response to the following HRS survey question, which is asked in all waves: [Q1] Do you currently have any life insurance? For whether or not an individual obtained a policy since the previous wave, we use the individual s response to the following HRS question, which is asked all waves starting in 1996: [Q2] Since (previous wave interview month-year) have you obtained any new life insurance policies? If the respondent answers yes, we consider him to have obtained a new policy. For whether or not an individual lapsed a policy since last wave, we use the individual s response to the following HRS question: [Q3] Since (previous wave interview month-year) have you allowed any life insurance policies to lapse or have any been cancelled? We also use the response to another survey question: [Q4] Was this lapse or cancellation something you chose to do, or was it done by the provider, your employer, or someone else? If the respondent answers yes to the first question and answers my decision to the second question, we consider him to have lapsed a policy. In the notation of the model we presented in the previous section, we construct an individual s period-t (or wave- t) decisions as follows: 6

8 For the individual who reported not having life insurance in the previous wave (d t 1 = 0), we let d t = 0 if the individual reports not having life insurance this wave; and d t = 1 if the individual reports having life insurance this wave ( yes to Q1). Because the individual does not own life insurance in period t 1 but does in period t, we interpret that he chose the optimal policy in period t given his state variables at t. For the individual who reported having life insurance in the previous wave (d t 1 > 1), we let d t = 0 if the individual reports not having life insurance this wave ( no to Q1). We let d t = 1 (i.e. the individual re-optimizes his life insurance) if the individual reports having life insurance this wave ( yes to Q1) and he obtained new life insurance policy ( yes to Q2), OR if the individual answered yes to Q1, reported lapsing (i.e. answered yes to Q3) and reported that lapse was his own decision (answered my own decision to Q4). Note that under this construction, we have interpreted the lapsing or obtaining of any policies as an indication that the respondent re-optimized his life insurance coverage. Finally, we let d t = 2 (i.e. he kept his previous life insurance policy unchanged) if the individual reports having life insurance this wave ( yes to Q1) and the individual did not report yes to obtaining new policy ( no to Q2) and the individual did not lapse any existing policy (either report no to Q3 or reported yes to Q3 but did not report my decision to Q4). Information About the Details of Life Insurance Holdings in the HRS Data. HRS also has questions regarding the face amount and premium payments for life insurance policies. However, there are several problems with incorporating these variables into our empirical analysis. First, the questions differ across waves. In the 1994 wave, questions were asked regarding total face amount and premium for term life policies; but for whole life policies only total face amount was collected. 6 From 2000 on, the HRS asked about the combined face value for all policies, combined face value for whole life policies, and the combined premium payments for whole life policies. Note that the premium for term life policies were not collected from 2000 on. Second, there is a very high incidence of missing data regarding life insurance premiums and face amounts. In our selected sample, 40.3% of our selected sample (1340 individuals) have at least one instance of missing face amount in waves when he reported owning life insurance. The incidence of missing values in premium payments is even higher. Third, even for those who reported face amount and premium payments for their life insurance policies, we do not know the choice set they faced when purchasing their policies. 6 The questions in 1994 wave related to premium and face amount are: [W6768]. About how much do you pay for (this term insurance/these term insurance policies) each month or year? [W6769]. Was that per month, year, or what? [W6770]. What is the current face value of all the term insurance policies that you have? [W6773]. What is the current face value of (this [whole life] policy/these [whole life] policies?) 7

9 For these reasons, we decide to only model the individuals life insurance decisions regarding whether to re-optimize, lapse or maintaining an existing policy, and only use the observed information about the above decisions in estimating the model. 2.1 Descriptive Statistics Patterns of Life Insurance Coverage and its Transitions. Table 3 provides the life insurance coverage and patterns of transition between coverage and no coverage in the HRS data. Panel A shows that among the 3,324 live respondents in 1996, 88.1% are covered by life insurance; among the 3,195 who survived to the 1998 wave, 85.7% owned life insurance, etc. Over the waves, the life insurance coverage rates among the live respondents seem to exhibit a declining trend, with the coverage rate among the 2,558 who survived to the 2006 wave being about 78.6%. Panel B and C show, however, that there are substantial transition between the coverage and no coverage. Panel B shows that among the 512 individuals who did not have life insurance coverage in 1994, almost a half (47.5%) obtained coverage in 1996; in later waves between 25.6% to 33.7% of individuals without life insurance in the previous wave ended up with coverage in the next wave. Panel C shows that there is also substantial lapsation among life insurance policyholders. In our data, between-wave lapsation rates range from 4.6% to 10.2%. Considering that our sample is relatively old and the tenures of holding life insurance policies in the HRS sample are also typically longer, these lapsation rates are in line with the industry lapsation rates reported in the introduction (see Table 1). Panel D shows that even among those individuals who own life insurance in both the previous wave and the current wave, a substantial fraction has changed the face amount of their coverage, or in the words of our model, re-optimized. Between 6.0% to 9.5% of the sample who have insurance coverage in adjacent waves reported changing the face amount of their coverages by their own decisions. Summary Statistics of State Variables, by Life Insurance Coverage Status. Table 4 summarizes the key state variables for the sample used in our empirical analysis. It shows that the average age of the live respondents in our sample is 61.1, which increases by less than two years in the next waves. This is as expected, because those who did not survive to the next wave because of death tend to be older than average. The mean of log household income in our sample is quite stable around to 10.73, with slight increase over the waves, possibly because low income individuals tend to die earlier. The next six rows report the mean of the incidence of health conditions, including high blood pressure, diabetes, cancer, lung disease, heart disease and stroke. It shows clear signs of health deterioration for the surviving samples over the years. The sum of the above 8

10 Table 3: Life Insurance Coverage and Transition Patterns in HRS: Wave Panel A: Life Insurance Coverage Status Currently covered by life insurance 2,927 2,739 2,524 2,313 2,187 2, % 85.7% 83.5% 81.0% 80.5% 78.6% No life insurance coverage % 14.3% 16.5% 19.0% 19.5% 21.4% Total live respondents 3,324 3,195 3,022 2,854 2,717 2,558 Panel B: Life Insurance Coverage Status Conditional on No Coverage in Previous Wave Life insurance coverage this wave % 33.4% 31.9% 33.7% 32.7% 25.6% No life insurance coverage this wave % 66.6% 68.1% 66.3% 67.3% 74.4% Total live respondents with no coverage last wave Panel C: Life Insurance Coverage Status Conditional on Coverage in Previous Wave Life insurance coverage this wave 2,684 2,614 2,394 2,163 2,024 1, % 92.7% 91.5% 89.8% 91.3% 90.9% No life insurance coverage this wave % 7.3% 8.5% 10.2% 8.7% 9.1% Total respondents with coverage last wave 2,812 2,821 2,615 2,409 2,218 2,078 Panel D: Whether Changed Coverage Amount Conditional on Coverage in Both Current and Previous Waves Did not change coverage amount 2,430 2,395 2,233 2,034 1,881 1, % 91.6% 93.3% 94.0% 92.9% 93.7% Changed coverage amount % 8.4% 6.7% 6.0% 7.1% 6.3% Total live respondents with coverage in both waves 2,684 2,614 2,394 2,163 2,024 1,888 9

11 Table 4: Summary Statistics of State Variables. Variable Description Wave Age of respondent (4.353) (4.343) (4.306) (4.285) (4.285) (4.250) Log household income (1.301) (1.208) (1.205) (1.204) (1.038) (0.915) Whether ever diagnosed with high blood pressure (0.490) (0.496) (0.499) (0.500) (0.495) (0.485) Whether ever diagnosed with diabetes (0.351) (0.366) (0.381) (0.405) (0.419) (0.435) Whether ever diagnosed with cancer (0.237) (0.273) (0.300) (0.335) (0.366) (0.390) Whether ever diagnosed with lung disease (0.257) (0.268) (0.271) (0.284) (0.312) (0.322) Whether ever diagnosed with heart disease (0.392) (0.410) (0.424) (0.442) (0.460) (0.475) Whether ever diagnosed with stroke (0.217) (0.234) (0.247) (0.257) (0.276) (0.298) Whether ever diagnosed with psychological problem (0.238) (0.258) (0.270) (0.283) (0.291) (0.301) Whether ever diagnosed with arthritis (0.488) (0.497) (0.500) (0.499) (0.494) (0.485) Sum of above conditions (1.241) (1.292) (1.311) (1.337) (1.395) (1.421) Whether married (0.357) (0.364) (0.363) (0.363) (0.363) (0.374) # of live respondents 3,324 3,195 3,022 2,854 2,717 2,558 Note: Standard deviations are in parenthesis. 10

12 Table 5: Summary Statistics of State Variables Conditional on Having Life Insurance Variable Description Wave Age of respondent (4.339) (4.323) (4.318) (4.284) (4.311) (4.260) Log household income (1.156) (1.081) (1.090) (1.130) (0.947) (0.857) Whether ever diagnosed with high blood pressure (0.490) (0.496) (0.500) (0.500) (0.495) (0.487) Whether ever diagnosed with diabetes (0.339) (0.355) (0.368) (0.395) (0.412) (0.427) Whether ever diagnosed with cancer (0.236) (0.270) (0.301) (0.341) (0.370) (0.393) Whether ever diagnosed with lung disease (0.253) (0.263) (0.267) (0.281) (0.305) (0.318) Whether ever diagnosed with heart disease (0.388) (0.408) (0.423) (0.442) (0.461) (0.475) Whether ever diagnosed with stroke (0.203) (0.222) (0.236) (0.246) (0.265) (0.288) Whether ever diagnosed with psychological problem (0.218) (0.243) (0.250) (0.267) (0.281) (0.289) Whether ever diagnosed with arthritis (0.488) (0.496) (0.500) (0.499) (0.493) (0.485) Sum of above conditions (1.206) (1.261) (1.300) (1.332) (1.377) (1.395) Whether married (0.334) (0.343) (0.348) (0.348) (0.345) (0.361) # of live respondents with life insurance coverage 2,927 2,739 2,524 2,313 2,187 2,011 Note: Standard deviations are in parenthesis. 11

13 Table 6: Summary Statistics of State Variables Conditional on Not Having Life Insurance Variable Description Wave Age of respondent (4.448) (4.454) (4.219) (4.290) (4.175) (4.216) Log household income (1.905) (1.637) (1.591) (1.431) (1.316) (1.081) Whether ever diagnosed with high blood pressure (0.496) (0.498) (0.499) (0.500) (0.496) (0.480) Whether ever diagnosed with diabetes (0.419) (0.421) (0.432) (0.439) (0.446) (0.459) Whether ever diagnosed with cancer (0.248) (0.289) (0.298) (0.305) (0.349) (0.378) Whether ever diagnosed with lung disease (0.288) (0.299) (0.290) (0.298) (0.337) (0.336) Whether ever diagnosed with heart disease (0.422) (0.426) (0.428) (0.446) (0.460) (0.475) Whether ever diagnosed with stroke (0.294) (0.293) (0.295) (0.300) (0.315) (0.332) Whether ever diagnosed with psychological problem (0.343) (0.329) (0.348) (0.338) (0.328) (0.338) Whether ever diagnosed with arthritis (0.489) (0.500) (0.500) (0.500) (0.495) (0.488) Sum of above conditions (1.434) (1.438) (1.350) (1.357) (1.463) (1.507) Whether married (0.463) (0.451) (0.423) (0.413) (0.423) (0.413) # of live respondents without life insurance coverage Note: Standard deviations are in parenthesis. 12

14 six health conditions steadily increase from 1.37 in 1996 to 2.34 in Finally, the marital status of the surviving sample seems to be quite stable, with the fraction married being in the range of 83% to 85%. Tables 5 and 6 summarize the mean and standard deviation of the state variables by the life insurance coverage status. There does not seem to be much of a difference in ages between those with and without life insurance coverage, but the mean log household income is significantly higher for those with life insurance than those without. Moreover, those with life insurance tend to be healthier than those without life insurance, and they are much more likely to be married than those without. 3 An Empirical Model of Life Insurance Decisions In this section we present a discrete choice model of how individuals make life insurance decisions. We will later implement the dynamic structural model. Our model is simple, yet rich enough to capture the dynamic intuition behind the life insurance models of Hendel and Lizzeri (2003) and Fang and Kung (2010a). Time is discrete and indexed by t = 1, 2,... In the beginning of each period t, an individual i either has or does not have an existing life insurance policy. If the individual enters period t without an existing policy, then he chooses between not owning life insurance (d it = 0) or optimally purchasing a new policy (d it = 1). If the individual enters period t with an existing policy, then, besides the above two choices, he can additionally choose to keep his existing policy (d it = 2). If an individual who has life insurance in period t 1 decides not to own life insurance in period t, we interpret it as lapsation of coverage. As we describe in Section 2, the choice d it = 1 for an individual who previously owns a policy is interpreted more broadly: an individual is considered to have re-optimized his existing policy if he reported purchasing a new policy or choosing to lapse an existing policy. The key intrepretation for choice d it = 1 is that the individual re-optimized his life insurance holdings. Flow Payoffs from Choices. Now we describe an individual s payoffs from each of these choices. First, let x it X denote the vector of observable state variables of individual i in period t, and let z it Z denote the vector of unobservable state variables. 7 These characteristics include variables that affect the individual s preference for or cost to owning life insurance, such as income, health and bequest motives. We normalize the utility from not owning life insurance (i.e., d it = 0)to 0; 7 We present the model here assuming the presence of both the observed and unobserved state variables. In Section 5, we will also estimate a model with only observed state variables. In that case, we should simply ignore the unobserved state vector z it. 13

15 that is, u 0 (x it, z it ) = 0 for all (x it, z it ) X Z. (1) The utility from optimally purchasing a new policy in state (x it, z it ), i.e., d it = 1, regardless of whether he previously owned a life insurance policy, is assumed to be: u 1 (x it, z it ) + ε 1it, (2) where ε 1it is an idiosyncratic choice specific shock, drawn from to a type I extreme value distribution. In our empirical analysis, we will specify u 1 (x it, z it ) as a flexible polynomial of x it and z it. Now we consider the flow utility for an individual i entering period t with an existing policy which was last re-optimized at at period ˆt. That is, let ˆt = sup {s s < t, d is = 1}. Let (ˆx it, ˆz it ) = (x, z iˆt iˆt ) denote the state vector that i was in when he last re-optimized his life insurance. We assume that the flow utility individual i obtains from continuing an existing policy purchased when his state vector was (ˆx it, ˆz it ) is given by u 2 (x it, z it, ˆx it, ˆz it, ε 2it ) = u 1 (x it, z it ) c((x it, z it ), (ˆx it, ˆz it )) + ε 2it, (3) where c ( ) can be considered a sub-optimality penalty, which may also include adjustment costs (see discussion below in Section 3.1), that possibly depends on the distance (x it, z it ), (ˆx it, ˆz it ) between the current state (x it, z it ) and the state in which the existing policy was purchased (ˆx it, ˆz it ). The adjustment can be positive or negative, depending on the factors that have changed. For example, if the individual was married when he purchased the existing policy but is not married now, then, all other things equal, the adjustment is likely to be negative; he would have less incentive to keep the existing policy. On the other hand, if the individual s health has deteriorated substantially, then obtaining a new policy could be prohibitively costly, in which case the adjustment is likely to be positive; he would have more incentive to keep the existing policy which was purchased during a healthier state. To summarize, we model the life insurance choice as the decision to either: 1) hold no life insurance, 2) purchase a new insurance policy which is optimal for the current state, or 3) continue with an existing policy. By decomposing the ownership decision into continuation vs. re-optimzation, our model is able to capture the intuition that an individual who has suffered a negative shock to a factor that positively affects life insurance ownership (such as income or bequest motive) may still be likely to keep his insurance if the policy was initially purchased a long time ago during a better health state. 14

16 Moreover, the decomposition of the ownership decision allows us to examine two separate motives for lapsation: lapsation because the individual no longer needs any life insurance, and lapsation because the policyholder s personal situation, i.e. (x it, z it ), has changed such that new coverage terms are required. Parametric Assumptions on u 1 and c Functions. In our empirical implementation of the model, we let the observed state vector x it include age, log household income, sum of the number of health conditions and marital status, and we let the unobserved state vector z it include z 1it, z 2it and z 3it which respectively represent the unobserved components of income, health and bequest motives. 8 In Section 6 below, we will describe how we anchor these unobservables to their intended interpretations and how we use sequential Monte Carlo method to simulate their posterior distributions. The primitives of our model are thus given by the utility function of optimally purchasing life insurance u 1, and the sub-optimality adjustment function c. Our specification for u 1 (x it, z it ) in our empirical analysis is given by u 1 (x it, z it ) = θ 0 + θ 1 AGE it + θ 2 (LOGINCOME it + z 1it ) and our specification for c( (x it, z it ), (ˆx it, ˆz it ) ) is: +θ 3 (CONDITIONS it + z 2it ) + θ 4 (MARRIED it + z 3it ), (4) c((x it, z it ), (ˆx it, ˆz it )) = θ 5 + θ 6 (AGE it AGE ˆ it ) + θ 7 (AGE it ÂGE it ) 2 ) ) 2 +θ 8 (LOGINCOME it + z 1it LOGINCOMEit ˆ ˆz 1it + θ 9 (LOGINCOME it + z 1it LOGINCOMEit ˆ ˆz 1it ) ) 2 +θ 10 (CONDITIONS it + z 2it CONDITIONS ˆ it ˆz 2it + θ 11 (CONDITIONS it + z 2it CONDITIONS ˆ it ˆz 2it MARRIEDit +θ 12 + z 3it MARRIED ˆ it ˆz 3it. (5) Transitions of the State Variables. The state variables which the an individual must keep track of depend on whether the individual is currently a policyholder. If he currently does not own a policy, his state variable is simply his current state vector (x it, z it ) ; if he currently owns a policy, then his state variables include both his current state vector (x it, z it ) and the state vector (ˆx it, ˆz it ) at 8 We do not include information regarding children also as potentially determinants of bequest motive for the following reasons. If we include the number of children as one of the state variables, there is pratically no variation in our data set due to the ages of the individuals in our sample. Thus the effect of the variable number of children will be soaked into the constant term. However, if we include age of the youngest child or number of dependent children (i.e. children below age 18), we would have to include the ages of all children as part of the state variables, which significantly increases the dimensionality of our problem and make it untractible computationally. 15

17 which he purchased the policy he currently owns. In our empirical analysis, we assume that the current state vectors (x it, z it ) evolve exogenously (i.e., not affected by the individual s decision) according to a joint distribution given by (x it+1, z it+1 ) f (x it+1, z it+1 x it, z it ). In particular, for the observed state vector x it, which includes age, log household income, sum of the number of health conditions and marital status, we estimate their evolution directly from the data. For the unobserved state vector z it, we will use sequential Monte Carlo methods to simulate its evolution (see Section 6.2 below for details). The evolution of the state vector (ˆx it, ˆz it ) is endogenous, and it is given as follows. If the individual does not own life insurance at period t, which we denote by setting (ˆx it, ˆz it ) =, then (x it, z it ) if d it = 1 (ˆx it+1, ˆz it+1 ) (ˆx it, ˆz it ) = = if d it = 0 (6) where denotes that the individual remains with no life insurance. If the individual owns life insurance at period t purchased at state (ˆx it, ˆz it ), then if d it = 0 (ˆx it+1, ˆz it+1 ) (ˆx it, ˆz it ) = = (x it, z it ) if d it = 1 (ˆx it, ˆz it ) if d it = 2. (7) 3.1 Discussion Dynamic Discrete Choice Model without the Knowledge of the Choice and Choice Set. As we mentioned in Section 2, we do not have complete information about the exact life insurance policies owned by the individuals, and for those whose life insurance policy we do know about, we do not know the choice set from which they choose. However, we do know whether an individual has re-optimized his life insurance policy holdings (i.e., purchase a new life insurance policy, or changed the amount of his existing coverage), or has dropped coverage etc. The data restrictions we face are fairly typically for many applications. 9 For example, in the 9 McFadden (1978) and Fox (2007) studied problems where the researcher only observes the choices of decisionmakers from a subset of choices. McFadden (1978) showed that in a class of discrete-choice models where choice specific error terms have a block additive generalized extreme value (GEV) distributions, the standard maximum likelihood estimator remains consistent. Fox (2007) proposed using semiparametric multinomial maximum-score estimator when estimation uses data on a subset of the choices available to agents in the data-generating process, thus relaxing the 16

18 study of housing market, it is possible that all we observe is whether a family moved to a new house, remained in the same house, or decided to rent; but we may not observe the characteristics (including purchase price) of the new house the family moved into, or the characteristics of the house/apartment the family rented. Our formulation provides an indirect utility approach to deal with such data limitations. Suppose that when individual i s state vector is (x it, z it ), he has a choice set L (x it, z it ) denoting all the possible life insurance policies that he could choose from. Note that L (x it, z it ) depends on i s state vector (x it, z it ), which captures the notion that life insurance premium and face amount typically depend on at least some of the characteristics of the insured. Let l L (x it, z it ) denote one such available policy. Let u (l; x it, z it ) denote individual i s flow utility from purchasing policy l. If he were to choose to own a life insurance, his choice of the life insurance contract from his available choice set will be determined by the solution to the following problem: V (x it, z it ) = max l L(x it,z it ) {u (l; x it, z it ) + ε 1it + βe [V (x it+1, z it+1 ) l, x it, z it ]}. (8) Let l (x it, z it ) denote the solution. Then the flow utility u 1 (x it, z it ) we specified in (2) can be interpreted as the indirect flow utility, i.e., u 1 (x it, z it ) = u (l (x it, z it ) ; x it, z it ). (9) It should be pointed out that, under the above indirect flow utility interpretation of u 1 (x it, z it ), in order for the error term ε 1it in (2) to be distributed as i.i.d extreme value as assumed, we need to make the assumption that ε it in (8) does not vary across l L (x it, z it ). This seems to be a plausible assumption. Interpretations of the Sub-Optimality Penalty Function c ( ). The sub-optimality penalty function c ( ) we introduced in (3) admits a potential interpretation that changing an existing life insurance policy may incur adjustment costs. To see this, consider an individual whose current state vector is (x it, z it ) and owns a life insurance policy he purchased at ˆt when his state vector was (x, z iˆt iˆt ). Suppose that he decides to change (lapse or modify) his current policy and re-optimize, but there is an adjustment cost of κ > 0 for changing. Thus, the flow utility from lapsing into no coverage for this individual will be u 0 (x it, z it ) = κ. distributional assumptions on the error term required for McFadden (1978). 17

19 The flow utility from re-optimizing, using the notation from (9), will be u 1 (x it, z it ) κ = u (l (x it, z it ) ; x it, z it ) κ. And the flow utility from keeping the existing policy is u 2 (x it, z it, ˆx it, ˆz it ) = u (l (ˆx it, ˆz it ) ; x it, z it ) = u (l (x it, z it ) ; x it, z it ) sub-optimality penalty { }} { [u (l (x it, z it ) ; x it, z it ) u (l (ˆx it, ˆz it ) ; x it, z it )]. (10) where we used the notation l (ˆx it, ˆz it ) to denote the optimal policy for state vector (ˆx it, ˆz it ). Note that the term [u (l (x it, z it ) ; x it, z it ) u (l (ˆx it, ˆz it ) ; x it, z it )] indeed measures the utility loss from holding a policy l (ˆx it, ˆz it ) that was optimal for state vector (ˆx it, ˆz it ), but sub-optimal when state vector is (x it, z it ). If we were to normalize the flow utility from not owning life insurance u 0 (x it, z it ) = 0, we essentially have c (x it, z it ; x iˆt, z iˆt ) = [u (l (x it, z it ) ; x it, z it ) u (l (ˆx it, ˆz it ) ; x it, z it )] κ. That is, our sub-optimality penalty c ( ) indeed incorporates the adjustment cost κ as a component. It should be clear from the above discussion that the adjust cost κ could easily be made a function of both (x it, z it ) and (x, z iˆt iˆt ). We will not be able to separate the sub-optimality penalty [u (l (x, z iˆt iˆt ) ; x it, z it ) u (l (x it, z it ) ; x it, z it )] from κ (x it, z it ; x, z iˆt iˆt ). Given the presence of adjust cost κ, we would expect that an existing policyholders will hold on to his policy until the sub-optimality penalty [u (l (x iˆt, z iˆt ) ; x it, z it ) u (l (x it, z it ) ; x it, z it )] exceeds the adjustment cost κ, if we ignore decisions driven by i.i.d preference shocks ε 1it and ε 2it. Limitations of the Indirect Flow Utility Approach. While the indirect flow utility approach we adopted in this paper to deal with the lack of information regarding individuals actual choices of life insurance policies and their relevant choice set is useful for our purpose of understanding why policyholders lapse their coverage (as we will demonstrate later), it has a major limitation. The indirect flow utilities u 1 (x it, z it ) and u 2 (x it, z it, ˆx it, ˆz it ) defined in (9) and (10) respectively, are derived only under the existing life insurance market structure. As a result, the estimated indirect flow utility functions are not primitives that are invariant to counterfactual policy changes that may affect the equilibrium of the life insurance market. Of course, this limitation is also present in other dynamic discrete choice models where the flow utility functions can have the interpretation 18

20 Table 7: Reduced-Form Logit Regression on the Probability of Buying Life Insurance, Conditional on Having No Life Insurance in the Previous Wave Variable Coefficient Std. Error Constant Age Logincome Number of Health Conditions Married Observations 2,717 Log likelihood -1,721.3 significant at 5% level significant at 1% level as reduced-form, indirect utility function of a more detailed choice problem Preliminary Results Before we estimate the dynamic structural model, we here present some preliminary empirical results documenting the relationship between life insurance coverage and observed state variables, such as age, log household income, number of health conditions and marital status, and their changes. We first present the reduced-form, Logit regressions, that examine the determinants of life insurance decisions; we then estimate a static version of the dynamic model we presented in Section 3 (i.e., assuming that the discount factor β = 0). 4.1 Reduced-Form Determinants of the Life Insurance Decisions Table 7 presents the coefficient estimates of a Logit regression on the probability of purchasing life insurance among those who did not have coverage in the previous wave. It shows that the richer, younger, healthier and married individuals are more likely to purchase life insurance coverage than the poorer, older, unhealthier and widowed individuals. Table 8 presents the estimates of a multinomial Logit regression for the probability of lapsing, changing coverage, or 10 For example, in many I.O. papers a reduced-form flow profit function is assumed. 19